The Stacks project

Lemma 10.78.6. Let $R \to S$ be a flat local homomorphism of local rings. Let $M$ be a finite $R$-module. Then $M$ is finite projective over $R$ if and only if $M \otimes _ R S$ is finite projective over $S$.

Proof. By Lemma 10.78.2 being finite projective over a local ring is the same thing as being finite free. Suppose that $M \otimes _ R S$ is a finite free $S$-module. Pick $x_1, \ldots , x_ r \in M$ whose images in $M/\mathfrak m_ RM$ form a basis over $\kappa (\mathfrak m)$. Then we see that $x_1 \otimes 1, \ldots , x_ r \otimes 1$ are a basis for $M \otimes _ R S$. This implies that the map $R^{\oplus r} \to M, (a_ i) \mapsto \sum a_ i x_ i$ becomes an isomorphism after tensoring with $S$. By faithful flatness of $R \to S$, see Lemma 10.39.17 we see that it is an isomorphism. $\square$


Comments (0)

There are also:

  • 4 comment(s) on Section 10.78: Finite projective modules

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 00O1. Beware of the difference between the letter 'O' and the digit '0'.